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Universality of Packing Dimension Estimates for Spectral Measures of Quasiperiodic Operators: Monotone Potentials

Netanel Levi

Abstract

Let $H$ be a quasiperiodic Schrödinger operator generated by a monotone potential, as defined in [16]. Following [20], we study the connection between the Lyapunov exponent $L\left(E\right)$, arithmetic properties of the frequency $α$, and certain fractal-dimensional properties of the spectral measures of $H$.

Universality of Packing Dimension Estimates for Spectral Measures of Quasiperiodic Operators: Monotone Potentials

Abstract

Let be a quasiperiodic Schrödinger operator generated by a monotone potential, as defined in [16]. Following [20], we study the connection between the Lyapunov exponent , arithmetic properties of the frequency , and certain fractal-dimensional properties of the spectral measures of .

Paper Structure

This paper contains 19 sections, 19 theorems, 41 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Continued fraction approximation and $\beta\left(\alpha\right)$
  4. Packing and Rényi dimension of sets and measures
  5. The packing dimension
  6. The Rényi dimension
  7. Basic concepts in the theory of discrete Schrödinger operators
  8. The Borel transform
  9. Half-line restrictions of Schrödinger operators
  10. Asymptotic properties of generalized eigenfunctions
  11. The Lyapunov exponent
  12. Green's function estimate for monotone potentials
  13. The setting and main results
  14. Green's function and decay estimates of solutions to (\ref{['ev_eq_line_op']})
  15. Green's function
  16. ...and 4 more sections

Key Result

Theorem 1.1

For every Borel set $A\subseteq\mathbb{R}$, let $\mu^x|_A=\mu^x\left(\cdot\cap A\right)$. For every $x\in\mathbb{R}$, we have the following.

Theorems & Definitions (36)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Lemma 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Remark 2.5
  • Proposition 2.6
  • ...and 26 more